Weather Derivative Volatility Surface Estimation

ABSTRACT

Systems and methods are provided for determining the volatility of weather derivative option contracts. Volatility levels are initially determined with conventional methods. Unreliable volatility levels are then replaced with futures contracts volatility levels. If the futures contracts volatility levels are not available or appear unreliable, meteorological volatility levels are utilized. The data may be reduced to a three dimensional surface and used when determining margin account requirements.

The present application is a continuation of U.S. patent applicationSer. No. 12/191,800 filed Aug. 14, 2008 and entitled “Weather DerivativeVolatility Surface Estimation,” the entire disclosure of which is herebyincorporated by reference.

FIELD OF THE INVENTION

Embodiments of the present invention relates to methods and systems fordetermining volatility of weather based financial instruments.

DESCRIPTION OF THE RELATED ART

Options contracts or options give their owners the right but not theobligation to buy, in the case of call options, or to sell, in the caseof put options, an underlying good, such as a company's stock or bond,at a specified “strike” price for a preset amount of time. When thepreset amount of time has lapsed, the option “expires.”

Exemplary options contracts include weather derivatives. Weatherderivatives include financial instruments that can be used byorganizations or individuals as part of a risk management to reduce riskassociated with adverse or unexpected weather conditions. Derivativecontracts based on heating degree days may be geared to how much below65 degrees Fahrenheit the temperature averages in a given city in agiven month. Derivative contracts based on monthly snowfall may begeared toward the amount of snowfall recorded in a given month in adesignated location. Other exemplary weather derivative contracts arelisted at the Chicago Mercantile Exchange and described on theexchange's website.

The volatility of options contracts can be an important factor whendetermining credit and liquidity exposure and setting marginrequirements for a clearing member or firm. Because of the nature ofweather derivatives, models used to determine the volatility of otheroptions contracts are often not accurate for determining the volatilityof weather derivatives. For example, it is not uncommon to expect largechanges in volatility of some contracts even as the maturity dateapproaches. In contrast, weather events often become more certain as thematurity date approaches.

There is a need in the art for improved systems and methods fordetermining the volatility of weather derivatives and setting marginrequirements.

SUMMARY OF THE INVENTION

Embodiments of the present invention overcome problems and limitationsof the prior art by providing systems and methods for determining thevolatility of weather derivative option contracts. Black Scholes orJewson models may be used to create initial volatility values.Unreliable volatility levels may be replaced with futures contractsvolatility levels. If the futures contracts volatility levels are notavailable or appear unreliable, meteorological volatility values areutilized. Meteorological volatility values may be determined fromhistorical and forecast meteorological data.

In one embodiment, a seasonally adjusted GARCH model is utilized. Thedata may be reduced to a three dimensional surface and used whendetermining margin account requirements.

In other embodiments, the present invention can be partially or whollyimplemented on a computer-readable medium, for example, by storingcomputer-executable instructions or modules, or by utilizingcomputer-readable data structures.

Of course, the methods and systems of the above-referenced embodimentsmay also include other additional elements, steps, computer-executableinstructions, or computer-readable data structures. In this regard,other embodiments are disclosed and claimed herein as well.

The details of these and other embodiments of the present invention areset forth in the accompanying drawings and the description below. Otherfeatures and advantages of the invention will be apparent from thedescription and drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention may take physical form in certain parts and steps,embodiments of which will be described in detail in the followingdescription and illustrated in the accompanying drawings that form apart hereof, wherein:

FIG. 1 shows a computer network system that may be used to implementaspects of the present invention;

FIG. 2 illustrates an implied volatility surface for a collection ofweather derivative option contracts;

FIG. 3 illustrates a method of determining volatility for a group ofweather derivative option contracts, in accordance with an embodiment ofthe invention; and

FIG. 4 illustrates an implied volatility surface for a collection ofweather derivative option contracts that has been modified with themethod shown in FIG. 3, in accordance with an embodiment of theinvention.

DETAILED DESCRIPTION OF THE INVENTION

Aspects of the present invention may be implemented with computerdevices and computer networks that allow users to perform calculationsand exchange information. An exemplary trading network environment forimplementing trading systems and methods is shown in FIG. 1. An exchangecomputer system 100 receives orders and transmits market data related toorders and trades to users. Exchange computer system 100 may beimplemented with one or more mainframe, desktop or other computers. Auser database 102 includes information identifying traders and otherusers of exchange computer system 100. Data may include user names andpasswords potentially with other information to identify users uniquelyor collectively. An account data module 104 may process accountinformation that may be used during trades. A match engine module 106 isincluded to match bid and offer prices. Match engine module 106 may beimplemented with software that executes one or more algorithms formatching bids and offers. A trade database 108 may be included to storeinformation identifying trades and descriptions of trades. Inparticular, a trade database may store information identifying the timethat a trade took place and the contract price. An order book module 110may be included to compute or otherwise determine current bid and offerprices. A market data module 112 may be included to collect market dataand prepare the data for transmission to users. A risk management module134 may be included to compute and determine a user's risk utilizationin relation to the user's defined risk thresholds. An order processingmodule 136 may be included to decompose variable defined derivativeproduct and aggregate order types for processing by order book module110 and match engine module 106.

The trading network environment shown in FIG. 1 includes computerdevices 114, 116, 118, 120 and 122. Each computer device includes acentral processor that controls the overall operation of the computerand a system bus that connects the central processor to one or moreconventional components, such as a network card or modem. Each computerdevice may also include a variety of interface units and drives forreading and writing data or files. Depending on the type of computerdevice, a user can interact with the computer with a keyboard, pointingdevice, microphone, pen device or other input device.

Computer device 114 is shown directly connected to exchange computersystem 100. Exchange computer system 100 and computer device 114 may beconnected via a T1 line, a common local area network (LAN) or othermechanism for connecting computer devices. Computer device 114 is shownconnected to a radio 132. The user of radio 132 may be a trader orexchange employee. The radio user may transmit orders or otherinformation to a user of computer device 114. The user of computerdevice 114 may then transmit the trade or other information to exchangecomputer system 100.

Computer devices 116 and 118 are coupled to a LAN 124. LAN 124 may haveone or more of the well-known LAN topologies and may use a variety ofdifferent protocols, such as Ethernet. Computers 116 and 118 maycommunicate with each other and other computers and devices connected toLAN 124. Computers and other devices may be connected to LAN 124 viatwisted pair wires, coaxial cable, fiber optics or other media.Alternatively, a wireless personal digital assistant device (PDA) 122may communicate with LAN 124 or the Internet 126 via radio waves. PDA122 may also communicate with exchange computer system 100 via aconventional wireless hub 128. As used herein, a PDA includes mobiletelephones and other wireless devices that communicate with a networkvia radio waves.

FIG. 1 also shows LAN 124 connected to the Internet 126. LAN 124 mayinclude a router to connect LAN 124 to the Internet 126. Computer device120 is shown connected directly to the Internet 126. The connection maybe via a modem, DSL line, satellite dish or any other device forconnecting a computer device to the Internet.

One or more market makers 130 may maintain a market by providing bid andoffer prices for a derivative or security to exchange computer system100. Exchange computer system 100 may also exchange information withother trade engines, such as trade engine 138. One skilled in the artwill appreciate that numerous additional computers and systems may becoupled to exchange computer system 100. Such computers and systems mayinclude clearing, regulatory and fee systems. Coupling can be direct asdescribed or any other method described herein.

The operations of computer devices and systems shown in FIG. 1 may becontrolled by computer-executable instructions stored on acomputer-readable medium. For example, computer device 116 may includecomputer-executable instructions for receiving order information from auser and transmitting that order information to exchange computer system100. In another example, computer device 118 may includecomputer-executable instructions for receiving market data from exchangecomputer system 100 and displaying that information to a user.

Of course, numerous additional servers, computers, handheld devices,personal digital assistants, telephones and other devices may also beconnected to exchange computer system 100. Moreover, one skilled in theart will appreciate that the topology shown in FIG. 1 is merely anexample and that the components shown in FIG. 1 may be connected bynumerous alternative topologies.

FIG. 2 illustrates an implied volatility surface 200 for a collection ofweather derivative option contracts. A strike price axis 202 representsstrike prices. A days to maturity axis 204 represents scaled days tomaturity and volatility axis 206 represents volatility of sigma. Surface200 may be created with a conventional volatility model, such as a BlackScholes or Jewson model. Option prices for weather derivatives that arefarther out of the money and for contracts having maturity dates furtheraway can be somewhat erratic and may not result in accurate volatilitylevels when used with existing models. For example, region 208represents spikes in the volatility of weather derivatives havingrelatively far away maturity dates.

FIG. 3 illustrates a method that may be used to determine volatility inaccordance with an embodiment of the invention. First, in step 302volatility levels for weather derivative option contracts having a rangeof strike prices and time to maturity are determined using an optionvolatility model. In various embodiments of the invention volatility maybe calculated via a Black Scholes or Jewson model from weekly weathercalls/puts. Calculating implied volatility for pricing puts and callsfrom a Black Scholes model can be done as follows:

P(S, T) = K ^(−rT)Φ(−d₂) − S Φ(−d₁).C(S, T) = S Φ(d₁) − K ^(−rT)Φ(d₂)where$d_{1} = \frac{{\ln \left( {S/K} \right)} + {\left( {r + {\sigma^{2}/2}} \right)T}}{\sigma \sqrt{T}}$$d_{2} = {\frac{{\ln \left( {S/K} \right)} + {\left( {r - {\sigma^{2}/2}} \right)T}}{\sigma \sqrt{T}} = {d_{1} - {\sigma {\sqrt{T}.{Here}}\mspace{14mu} \Phi \mspace{14mu} {is}\mspace{14mu} {the}\mspace{14mu} {standard}\mspace{14mu} {normal}\mspace{14mu} {cumulative}\mspace{14mu} {distribution}}}}$function.

where: P=put price, C=call price, S=underlying asset price, K=strikeprice, r=risk free rate, sigma=implied volatility and T=remaining timeto maturity/expiration.

If we know P or C since the market determined the price, intra-day or atclose of business we also know: S, K, r and T. Those values can beplugged in and a search performed to find the closest sigma thatproduces the realized P or C. The search algorithm can be any of manywidely used algorithms such as the bisection method or Newton-Raphsonmethod.

Next, in step 304 it is determined whether a volatility level deviatesfrom adjoining volatility levels by a predetermined threshold. Such adeviation may result from insufficient or unreliable data. Region 208(shown in FIG. 2) corresponds to volatility levels that deviate fromadjoining volatility levels by a threshold. One skilled in the art willappreciate that different thresholds may be chosen for differentvolatility surfaces and purposes. When a deviation is found, next instep 306 it is determined whether a corresponding futures contractvolatility level is available that will not exceed adjoining volatilitylevels by the predetermined threshold. When a futures contractvolatility level is available, in step 308 the options contractvolatility level is replaced with a futures contract volatility level.

When a suitable futures contract volatility level is not available, instep 310 meteorological volatility is determined from historical andforecast meteorological data. Price volatility is related to weathervolatility for weather derivative contracts. As time to maturityapproaches various combinations of actual and forecasted data are used.For example, when considering a ten day contract based on temperature,at 10 days to maturity forecasted data issued. At one day to maturitynine days of actual data and one day of forecasted data will be used. Asthe maturity date approaches more actual and less forecasted data isused so volatility tends to decline.

In one embodiment, a GARCH model may be used to forecast volatility upto 30 days annualized that correspond to the weather derivative optioncontract volatilities determined in step 302. An exemplary GARCH modelis as follows:

GARCH (1,1)

σ_(t) ²=α₀+α₁α_(t-1) ²+β₁σ_(t-1) ²,

where

α_(t)=σ_(t)ε_(t),

α₀>0, α_(i)≧0, β_(j)≧0,

(α_(i)+β_(i)<1.

and eplison_t is a sequence of independent and identically distributedrandom variables with mean 0 and variance 1.

This GARCH model could also be used to find the volatility across across sectional slice of the volatility surface where by the volatilityof the strike or strike/current price (for puts) or current price/strike(for calls) time series could be used to forecast forward a futurevolatility level for that strike or stike/current price (for puts) orcurrent price/strike (for calls). As such, any forward missingvolatility term points in the surface will be filled. A smoothingalgorithm could be used such that the forward forecast may span (10 to30 days) and the average of the forward volatiles could be taken as theforecasted volatility for that strike or stike/current price (for puts)or current price/strike (for calls). Note that the relationship betweenputs and calls could be valued via an arbitrage free assumptions calledput-call parity based on the equation below:

(C(t)+K·(t,T)=P(t)+S(t)

where

-   -   C(t) is the value of the call at time t,    -   P(t) is the value of the put,    -   S(t) is the value of the underlying weather future,    -   K is the strike price, and    -   B(t, T) value of a treasury/risk free security that matures at        time T.

In various embodiments the GARCH model may be adjusted for seasonality.To adjust for seasonality a year may be divided into quarters and it maybe assumed that a stable relationship exists across the same quarter sothat a better forecast can be obtained by using the current quarter'svolatility as compared to the last year's same quarter. To do this,instead of regressing the current volatility against yesterday's (t−1)volatility is regressed against the volatility (or average volatilityover say 10, 20 or 30 days depending on which gives a more statisticallysignificant and stable relationship) at the same time but last year.Thus:

σ_(t) ²=(α_(0,1)+α_(1,1)α_(t-1) ²+β_(1,1)σ_(t-1) ²)*s₁+(α_(0,2)+α_(1,2)α_(t-2) ²+β_(1,2)σ_(t-2) ²)*s₂+(α_(0,3)+α_(2,3)α_(t-3) ²+β_(1,3)σ_(t-3) ²)*s₃+(α_(0,4)+α_(2,4)+α_(t-4) ²+β_(1,4)σ_(t-4) ²)*s ₄

where

α_(t)=σ_(t)ε_(t),

-   -   t: represent today    -   t−1: represents the same time today, but last year and so on to        t−2,3,4    -   s1, s2, s3, s4: are seasonality switches such that when one of        them is 1 the others are zero (for example if month is January,        February or March then s1=1 and s2=s3=s4=0 and so on)    -   Sigma_t−1: may be an average or a spot volatility.    -   Alphas and betas: are estimated via regression.

Next, in step 312 the options contract volatility level is replaced withthe meteorological volatility level. After step 312, control is returnedto step 304 and the loop is repeated until there are no volatilitylevels that deviate from adjoining volatility levels by thepredetermined threshold.

Interpolation techniques may then be used to create a solid surface. Forexample, a fifth or sixth degree polynomial that fits the discretevolatility points perfectly and would be able to forecast intermediatepoints. This is an example of an n-degree polynomial where f(x) would bevolatility and x is time. This may be repeated for volatiles across timewith constant strike prices

f(x)=a _(n) x ^(n) +a _(n-1) x ^(n-1) + . . . +a ₂ x ² +a ₁ x+a ₀

In an alternative embodiment, x is strike price and time is keptconstant and f(x) represents implied volatility. If prices were usedthen fit a polynomial to the price series, then the forecastedintermediate price may be used to reverse out an implied volatilitynumber via Black Scholes or Jewson models. Other interpolationtechniques can be used instead of polynomial fitting, such as quadraticor cubic splines or linear interpolation.

In step 314 an amount of credit risk available to a trader or otherentity may be calculated using a portfolio risk management determinationmethod and the derived volatility levels. An exemplary portfolio riskmanagement determination method is the Standard Portfolio Analysis ofRisk (SPAN®) method. The Standard Portfolio Analysis of Risk (SPAN®)method was developed by the Chicago Mercantile Exchange for calculatingperformance bond requirements. Risk management analysis may be performedacross multiple financial instruments at multiple exchanges includingpending orders at the multiple exchanges. In one implementation, aclearing firm may set a predetermined risk threshold for a tradingentity and use the Standard Portfolio Analysis of Risk (SPAN®) method todetermine whether the new order would cause the trading entity to exceedthe predetermined threshold. The predetermined threshold may be dynamicand based in part on conditions external to the trading entity's orders,conditions external to financial instrument and/or conditions at anexchange other than the exchange that received the new order. Thethreshold may be periodically recalculated by the entity that receivedthe new order.

The Standard Portfolio Analysis of Risk (SPAN®) method or otherportfolio risk management determination methods may calculate a higherrisk level when the new order is a buy order for a financial instrumentthat has a high correlation to an existing buy order for a differentfinancial instrument. In various embodiments the portfolio riskmanagement determination method may determine a risk associated with allbuy and/or sell orders being matched. In other embodiments of theinvention, the portfolio risk management determination method maydetermine the risk associated with a subset of all buy and/or sellorders being matched. The portfolio risk management determination methodmay be used to set margin account requirements. A margin accountrequirement is the money that a trader must deposit into his or hertrading account in order to trade options.

The amount of credit or risk available and/or any of the volatility datamay be displayed on a display device in step 316. FIG. 4 illustrates animplied volatility surface 400 for a collection of weather derivativeoption contracts that has been modified with the method shown in FIG. 3.For example, the spikes in region 208 (shown in FIG. 2) have beeneliminated. Of course, in some embodiments the data may be used in oneor more other processes and may not be displayed to users.

The present invention has been described herein with reference tospecific exemplary embodiments thereof. It will be apparent to thoseskilled in the art, that a person understanding this invention mayconceive of changes or other embodiments or variations, which utilizethe principles of this invention without departing from the broaderspirit and scope of the invention as set forth in the appended claims.All are considered within the sphere, spirit, and scope of theinvention.

1. A method of determining a volatility for a group of weatherderivative products; the method comprising: (a) determining at aprocessor volatility levels for weather derivative option contractshaving a range of strike prices and a range of times to maturity usingan option volatility model; (b) identifying at the processor any of theweather derivative option contract volatility levels that deviate fromadjoining volatility levels corresponding to weather derivative optioncontracts by a predetermined threshold; (c) replacing the volatilitylevels identified in (b) with weather derivative futures contractsvolatility levels for corresponding futures contracts or meteorologicalvolatility levels; and (d) displaying an implied volatility surface. 2.The method of claim 1, wherein the option volatility model comprises aBlack Scholes model.
 3. The method of claim 1, wherein the optionvolatility model comprises a Jewson model.
 4. The method of claim 1,wherein the implied volatility surface includes volatility levels for atleast some of the weather derivative option contracts and at leasteither (i) a volatility level for a weather derivative futures contractfrom (c) or (ii) a meteorological volatility level.
 5. The method ofclaim 1, where (c) comprises determining meteorological volatility usinga seasonally adjusted GARCH model.
 6. A method of determining a marginaccount requirement for a portfolio that includes at least one weatherderivative product, the method comprising: (a) determining at aprocessor volatility levels for weather derivative contracts having arange of strike prices and a range of times to maturity using an optionvolatility model; (b) identifying at the processor any of the weatherderivative option contract volatility levels that deviate from adjoiningvolatility levels corresponding to weather derivative option contractsby a predetermined threshold; (c) replacing the volatility levelsidentified in (b) with futures contracts volatility levels forcorresponding futures contracts or with meteorological volatility; and(d) calculating the amount of credit or risk available using a portfoliorisk management determination method and at least one of the volatilitylevels.
 7. The method of claim 6, wherein the portfolio risk managementdetermination method comprises the Standard Portfolio Analysis of Riskmethod.
 8. The method of claim 6, wherein the option volatility modelcomprises a Black Scholes model.
 9. The method of claim 6, wherein theoption volatility model comprises a Jewson model.
 10. The method ofclaim 6, further including: (e) displaying an implied volatility surfacethat includes volatility levels for at least some of the weatherderivative option contracts and a volatility level for a weatherderivative futures contract.
 11. The method of claim 6, where (c)comprises determining meteorological volatility using a seasonallyadjusted GARCH model.
 12. A non-transitory computer-readable storagemedium containing computer-executable instructions for performing thesteps comprising: (a) determining at a processor volatility levels forweather derivative contracts having a range of strike prices and a rangeof times to maturity using an option volatility model; (b) identifyingweather derivative contract volatility levels that deviate fromadjoining volatility levels corresponding to weather derivative optioncontracts by a predetermined threshold; and (c) replacing the volatilitylevels identified in (b) with futures contracts volatility levels forcorresponding futures contracts or with meteorological volatility. 13.The computer-readable medium of claim 12, further includingcomputer-executable instructions for performing: (d) generating athree-dimensional display of the volatility levels.
 14. Thecomputer-readable medium of claim 12, wherein the option volatilitymodel comprises a Black Scholes model.
 15. The computer-readable mediumof claim 12, wherein the option volatility model comprises a Jewsonmodel.
 16. The computer-readable medium of claim 13, wherein thethree-dimensional display comprises an implied volatility surface. 17.The computer-readable medium of claim 12, further includingcomputer-executable instructions for performing: (d) calculating theamount of credit or risk available using a portfolio risk managementdetermination method and at least one of the volatility levels.
 18. Thecomputer-readable medium of claim 17, wherein the portfolio riskmanagement determination method comprises the Standard PortfolioAnalysis of Risk method.
 19. A method comprising: (a) determining at aprocessor volatility levels for weather derivative option contractshaving a range of strike prices and a range of times to maturity usingan option volatility model; (b) identifying at the processor unreliablevolatility levels from the volatility levels determined in (a); (c) ifreliable futures contracts volatility levels are available, replacingthe unreliable volatility levels with corresponding futures contractsvolatility levels; and (d) if reliable futures contracts volatilitylevels are not available, replacing the unreliable volatility levelswith corresponding meteorological volatility levels.
 20. The method ofclaim 19, further including: (e) calculating an amount of credit or riskavailable using a portfolio risk management determination method and atleast one of the volatility levels.